Bayesian optimization based on the Gaussian process upper confidence bound (GP-UCB) offers a theoretical guarantee for optimizing black-box functions. In practice, however, black-box functions often involve input uncertainty. To handle such cases, GP-UCB can be extended to optimize evaluation criteria known as robustness measures. However, GP-UCB-based methods for robustness measures require a trade-off parameter, $β$, which, as in the original GP-UCB, must be set sufficiently large to ensure theoretical validity. In this study, we propose randomized robustness measure GP-UCB (RRGP-UCB), a novel method that samples $β$ from a chi-squared-based probability distribution. This approach eliminates the need to explicitly specify $β$. Notably, the expected value of $β$ under this distribution is not excessively large. Furthermore, we show that RRGP-UCB provides tight bounds on the expected regret between the optimal and estimated solutions. Numerical experiments demonstrate the effectiveness of the proposed method.